In mathematics, a pointed set is a set with a distinguished element, which is called the basepoint. Maps of pointed sets (based maps) are those functions that map one basepoint to another, i.e. a map such that . This is usually denoted
- .
Pointed sets may be regarded as a rather simple algebraic structure. In the sense of universal algebra, they are structures with a single nullary operation which picks out the basepoint.
The class of all pointed sets together with the class of all based maps form a category.
A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with one element.
Famous quotes containing the words pointed and/or set:
“See! those fiendish lineaments graven on the darkness, the writhed lip of scorn, the mockery of that living eye, the pointed finger, touching the sore place in your heart! Do you remember any act of enormous folly, at which you would blush, even in the remotest cavern of the earth? Then recognize your Shame.”
—Nathaniel Hawthorne (1804–1864)
“A more secret, sweet, and overpowering beauty appears to man when his heart and mind open to the sentiment of virtue. Then he is instructed in what is set above him. He learns that his being is without bound; that to the good, to the perfect, he is born, low as he now lies in evil and weakness.”
—Ralph Waldo Emerson (1803–1882)